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Paraconsistent logics drop the notion of global consistency, instead they have a notion of local consistency.

In sheaf-theory, or categorical logic, as in topos theory, there is a notion of local truth.

Is it possible to think of paraconsistent logics in terms of topos theory - that is locally?

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  • Would it be logically equivalent to your question, to ask what notion of locality applies here? I suppose that there is some topology involved: if so, how is that topology defined? – Niel de Beaudrap Apr 23 '14 at 6:53
  • I'm not sure if its logically equivalent, but it is important; its the notion of a site that generalises the notion of a topological space; it does this by axiomatising the properties that characterises not the topology, but the category of open sets OX of a space X - thats the notion of a coverage – Mozibur Ullah Apr 23 '14 at 7:51
  • another way to think of it is that a sheaf of sets is equivalently an etale bundle, a bundle whose projection is a local homemorphism. This is the simplest example of a topoi, they generalise to Grothendieck topoi (this uses sites), and the equivalent formulation as a bundle becomes a fibration - so we have unique path lifting (upto a point in the cofibre); – Mozibur Ullah Apr 23 '14 at 8:01
  • In elementary topoi, generalise the former by axiomatising the notion of a subobject classifier, then a lawvere-tierney topology is a closure operator on the truth object - and this determines local truth there. – Mozibur Ullah Apr 23 '14 at 8:07
  • Stretching myself to quickly grasp the category theory here: the idea seems similar to that of an atlas (minus all of the properties specific to manifolds), and the idea of "locality" is essentially when all reasoning can be performed within a single "chart", is that about right? It would then seem that your question is whether or not there is a mode of paraconsistent reasoning, which has the property that all contradictory ensembles of propositions (e.g. pairs P and ¬P) cannot be contained within a single chart, i.e. one can prove that they cannot be accessed/used simultaneously. – Niel de Beaudrap Apr 23 '14 at 8:09
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This is far from an actual answer, but it suggests support for the idea.

If a construction were to proceed here, a few first steps would be:

  1. to find a meaningful example of a paraconsistent logic that allows us to think clearly about it and judge the success of our project as it proceeds
  2. to answer Niel de Beaudrap's concern as to whether there is a compelling topology on the space of its decisions
  3. to think about what kinds of mappings from that topology onto set-theory or measures seems to capture our 'degree of consistency'.

It seems to me that the primary model of paraconsistency that we use on a daily basis is morality. People who reason about morality in one very conventional way admit contradictions in their moral axioms all the time. Then they work outward on difficult issues from a highly conflicted position to one which achieves an acceptable level of consistency by taking in more and more perspective.

So a good model, in my head, would be a modal logic on the mode of "ought", with a common set of conflicting axioms, and degrees of consistency.

If you don't agree that morality works that way, Common Law explicitly tries to. Laws depend upon past application, and precedent matters more the closer it is or the more settled it is and the whole point of each judgement is to maintain or increase the internal consistency of the system, hoping that consistent and not offensive eventually devolves on just.

So I think there probably is a clearly articulated topology here, with open sets being something like 'must be considered to decide'. If you start from a very clear prescriptive cult doctrine and some Bayesian version of the 'ought' logic, or a rigidly statutory version of Common Law with an 'expert system' notion of an AI judge, you could probably present an approximation to that topology as simply as a graph.

It would not be a simple graph, and it would have to allow for degrees, regressions, etc., it may need probabilistic or bi-measured edges, etc. (to capture recency and settledness?) But, in concept, each statute or tenet applies to bodies of other decisions in a vaguely hierarchical way.

A second potential example might be causal flow or 'collapse' in a quantum environment limited by relativity or some other localizing principle. [Out comes the harp /] If you adopt my favorite notion of time as accumulated entropy, I can imagine making this a nice clean topology like a metric space on a manifold, by considering the merger of the local balances on entropy as the wavefronts sweep outward from separate decisions to find a global consensus of the direction in which entropy increases. The measure on consistency could be driven by "How long do I have before I am globally observable?"

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In his book Inconsistent mathematics, Chris Mortensen introduces complement-topoi and closed set sheaves - which I think do what you want. Unfortunately, there's some sloppiness in his presentation. Here's how I would choose to describe the situation:

  • We define "complement-topoi" and "closed set sheaves" and show (i) how complement-topoi can be built from "closed set sheaves" and (ii) how each complement-topos has an associated "internal paraconsistent logic."

  • We then observe - this is completely trivial - that complement-topoi and topoi coincide. This doesn't render the ideas above pointless of course. The "conservative" slogan at this point would be, "Every topos has an internal paraconsistent logic, "dual" in a precise sense to its usual internal intuitionistic logic;" a bit more radically we could say something like "starting from closed sets and paraconsistency rather than open sets and intuitionism, we arrive at the same collection of categories, so topoi can be equally thought of "intuitionism-y" or "paraconsistency-y.""

Mortensen, however, treats topoi and complement-topoi as genuinely different things by considering the witnessing data to be intrinsic to the category itself; see the discussion on page 105.

Let me end, however, on a positive note re: Mortensen's book. Personally, despite the sloppiness mentioned above (and some other issues), I find the book entirely worth reading if one approaches it with caution. In particular, the first two chapters really solidified my interest in paraconsistency and relevance logic.

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